The coordinates \(X^{\alpha x}\)  also establish a set of coordinate basis vectors \(\bar{\mathbf {e}}_{\alpha x}\)  on \(\Gamma \)  (see e.g., [1]}), given byAlthough it is unusual to think of basis vectors \(\bar{e}_{\alpha x}\)  as linear differential operators \(\delta /\delta X^{\alpha x}\) , the index notation makes it clear that they do, in fact, transform in exactly the same way. And, this is nonetheless a common convention, so we employ it here as well.
\(\bar{\mathbf {e}}_{\alpha x} = \frac{\delta }{\delta X^{\alpha x}} = \left(\frac{\delta }{\delta \rho (x)},\frac{\delta }{\delta \Phi (x)}\right).\) 
